Question

Does this infinite series have a closed form?

Answer 1

\[ \sum_{i=1}^{\infty} \frac{1}{p_i^{p_i}} \]

Where \(p_i\) is the ith prime.

Answer 2

It appears to approach this since the terms die off really quickly:

\[S \approx 0.287358...\]

Does that resemble anything you've seen before? All I can say is it's slightly larger than e/10. Hahaha.


http://www.wolframalpha.com/input/?i=1%2F2%5E2%2B1%2F3%5E3%2B1%2F5%5E5%2B...&t=crmtb01

Answer 3

Turns out base e and base 10 are more related than we thought? Ok probably not haha.

Answer 4

https://oeis.org/A094289

Answer 5

looks there isn't much online on this series

Answer 6

http://math.stackexchange.com/questions/1344000/what-is-known-about-the-sum-sum-frac1pp-of-reciprocals-of-primes-raised-to